1 ΔNCPV as Outcome Variable

“All baseline plaque metrics (coronary artery calcium, NCPV, total plaque score, and percent atheroma volume) were strongly associated with the change in NCPV.”

Change in noncalcified plaque volume \(\Delta \text{NCPV}\) was the outcome:

\[ \Delta \text{NCPV} = \text{NCPV}_{1} - \text{NCPV}_0 \]

They regressed \(\Delta \text{NCPV}\) directly on its baseline value \(\text{NCPV}_0\):

\[ \Delta \text{NCPV} = \alpha + \beta \, \text{NCPV}_0 + \varepsilon \]

But this introduces mathematical coupling, because \(\text{NCPV}_0\) appears on both sides of the equation:

\[ \text{NCPV}_{1} - \text{NCPV}_0 = \alpha + \beta \, \text{NCPV}_0 + \varepsilon \]

1 ΔNCPV as Outcome Variable

1 ΔNCPV as Outcome Variable

From Oldham* (1962): \(\beta > 0 \quad\text{if}\quad \rho > \frac{\sigma_0}{\sigma_1}\)

The slope depends on:

• correlation between baseline and follow-up (\(\rho\))
• relative spread of the two time points

So a positive or negative slope arises purely from the math.

In this study:

• Almost all participants had increased NCPV
• Implies \(\rho\) was moderate-to-high, and \(\sigma_1 > \sigma_0\)

These conditions tend to bias the slope upward; whether the resulting positive association reflects biology, coupling, or both cannot be determined from this model.

*Source: Oldham, 1962, J. Chronic Dis.

1 ΔNCPV as Outcome Variable

n <- 100 # Set the number of values to generate
ncpv_0 <- rnorm(n, mean = 100, sd = 10) # Create 100 random numbers centered around 100
ncpv_1 <- rnorm(n, mean = 110, sd = 10) # Create another 100 random numbers, centered around 110  
delta_ncpv <- ncpv_1 - ncpv_0 # Subtract the first set from the second to get the difference

1 ΔNCPV as Outcome Variable

An alternative is to model NCPV at follow-up (\(\text{NCPV}_1\)) directly while adjusting for baseline NCPV. This example uses ApoB as the independent variable:

\[ \text{NCPV}_1 = \alpha + \gamma\,\text{NCPV}_0 + \beta ApoB + \varepsilon \]

This approach avoids mathematical coupling, reduces residual variance, and allows the coefficient on \(ApoB\) to reflect biological association — not algebraic structure.

You could test whether baseline NCPV predicts follow-up using a mixed-effects model with a Time × Baseline interaction:

\[ \text{NCPV}_{ij} = \alpha + \gamma\,\text{Time}_{ij} + \beta\,\text{NCPV}_{0j} + \delta\,(\text{Time}_{ij} \cdot \text{NCPV}_{0j}) + b_j + \varepsilon_{ij} \]

but the with the number of subjects this must be done carefully.

1 ΔNCPV as Outcome Variable

Plaque begets plaque: biology or a mathematical artifact?

2 Univariable Linear Models

“Linear models on the primary (NCPV) and secondary outcomes were univariable”

Despite having multiple baseline predictors available
(age, sex, ApoB, Δ-ApoB, CAC, NCPV₀, LDL-C exposure),
each was tested separately in single-predictor regressions.

This modeling choice introduces omitted-variable bias:

“…omitting a relevant variable from a model which explains the independent and dependent variable leads to biased estimates.”
Wilms (2021)

When a predictor is correlated with both the outcome and another omitted variable, its coefficient may absorb the effect of that omitted factor.

2 Univariable Linear Models

Example: ΔNCPV, ApoB and Age

They modeled:

\[ \Delta \text{NCPV} = \alpha + \beta \cdot \text{ApoB} + \varepsilon \]

But if age also predicts ΔNCPV and correlates with ApoB, then \(\beta\) is biased — it partly reflects the effect of age.

A more appropriate model would be:

\[ \Delta \text{NCPV} = \alpha + \beta_1 \cdot \text{ApoB} + \beta_2 \cdot \text{Age} + \varepsilon \]

This separates the contribution of ApoB from that of age.

Univariable Linear Models are make confounding almost certain, especially in small, non-randomized human data, undermining any claim of association or non-association.

Univariable Linear Models are exploratory and can be misleading.

2 Univariable Linear Models

So they did use multivariable models! — but only on the follow-up NCPV, not for ΔNCPV, the paper’s main endpoint?

This selective use raises questions:

• Why not model ΔNCPV with adjustment for known confounders?
• Why apply different modeling rules depending on the predictor?
• Why include some adjusted models in the paper?

Were multivariable models used selectively for some reason? And why not on the study’s main endpoint?

3 Bayesian Inference

“Bayesian inference adds credence to finding that there is no association between NCPV vs LDL-C or ApoB…”

The authors used Bayesian inference to support their finding that ApoB has no association with plaque progression.

This is unusual in a non-randomized, uncontrolled, 1-year observational study on a highly restricted sample:

• No causal inference can be made from this study design
• It’s done on univariable, unadjusted models
• They use Bayes factors to imply (strongly) the absence of effect, not just lack of evidence

This is a misuse of Bayesian inference; to “add credence” to a claim of no effect, based on an uncontrolled study design and simplistic modeling.

4 Bayesian Prior Choice

“Bayes factors were calculated … using an ~ rscale = 0.8 value of 0.8 to contrast a moderately informative prior with a conservative distribution width (to allow for potential large effect sizes”

Bayes factors compare how well the data are predicted under H₀ vs. H₁ — but H₁ isn’t just any nonzero effect; it’s a distribution of plausible effect sizes defined by the prior.

The authors used a wide, Cauchy-like prior (r = 0.8), “due to the well-documented association between ApoB changes and coronary plaque changes.”

• The model expected large effects from ApoB, so small effects could be seen as evidence for no effect.

This choice of prior isn’t wrong, it is fragile, subjective, and drives the result.

5 Bayes Factor Interpretation

“In other words, these data suggest it is 6 to 10 times more likely that the hypothesis of no association between these variables (the null) is true as compared to the alternative.”

That is a overstatement of what the Bayes Factor tells us.

A Bayes factor of 6–10 means the data are 6–10× more likely under the null model than under the alternative model, not that the null hypothesis is 6–10× more likely to be true.

They could have said: “The data are 6–10 times more likely under the no-association model than under the alternative”

\[ \text{Posterior Odds} = \text{Bayes Factor} \times \text{Prior Odds} \]

To claim that the null is 6–10× more likely to be true, they would have to assume prior odds = 1:1 — and state that explicitly. They didn’t.

Source: Bayes Factors – Kass Raferty, 1995

5 Bayes Factor Interpretation

Even without other issues (e.g. confounding / non-adjusted variables, short follow-up, etc.), the reported BF of 6.3 for ΔNCPV ~ ApoB reflects only moderate evidence for no effect — not strong or decisive.

(Assuming BF₁₀ as per standard conventions; if BF₀₁, it reverses)

Summary

No Adjustment for Multiple Comparisons

Predictors tested against two outcomes (Δ-NCPV, Δ-TPS) → ≥10 comparisons

• Bonferroni: α = 0.05 → α′ ≈ 0.005
  Baseline CAC (P < 0.001) would survive; others might not
  
• Benjamini–Hochberg FDR maintains power with correlated tests

Numerous regressions were run — increasing false positive risk
But no multiple testing correction was applied.

Perhaps the authors viewed this as exploratory, where correction is often skipped —
but then why title the paper “Plaque Begets Plaque, ApoB Does Not”?

Zero-inflation, censoring, heteroscedasticity